3. The Equation for the terminal Side `theta` is `y=1/2x` and `x<0`. Find value of all six trigonometric functions
Solution:
`y=1/2x`
`y=x/2`
Here, `x<0`
Let `x=-2`
So `y=1/2xx-2=-1`
`P(-2,-1)`
Opposite side `(y)`, adjacent side `(x)` and hypotenuse `(r)`
`sin(theta), cos(theta), tan(theta)` fromula
`sin(theta) = "opposite"/"hypotenuse" = y/r`
`cos(theta) = "adjacent"/"hypotenuse" = x/r`
`tan(theta) = "opposite"/"adjacent" = y/x`
`csc(theta) = "hypotenuse"/"opposite" = r/y`
`sec(theta) = "hypotenuse"/"adjacent" = r/x`
`cot(theta) = "adjacent"/"opposite" = x/y`
Here `x=-2` and `y=-1`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=(-2)^2 + (-1)^2`
`=4 + 1`
`=5`
`:.r=sqrt(5)`
So, `x=-2,y=-1 and r=sqrt(5)`
`(1)` `sin(theta)=y/r=(-1)/(sqrt(5))=(-sqrt(5))/5=-0.44721`
`(2)` `cos(theta)=x/r=(-2)/(sqrt(5))=(-2sqrt(5))/5=-0.89443`
`(3)` `tan(theta)=y/x=(-1)/(-2)=1/2=1/2`
`(4)` `csc(theta)=r/y=(sqrt(5))/(-1)=-sqrt(5)=-2.23607`
`(5)` `sec(theta)=r/x=(sqrt(5))/(-2)=(-sqrt(5))/2=-1.11803`
`(6)` `cot(theta)=x/y=(-2)/(-1)=2`
This material is intended as a summary. Use your textbook for detail explanation.
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